Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $n \neq 0$. $p = \dfrac{n}{5n - 10} \div \dfrac{4n}{3(n - 2)} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{n}{5n - 10} \times \dfrac{3(n - 2)}{4n} $ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ n \times 3(n - 2) } { (5n - 10) \times 4n } $ $ p = \dfrac {n \times 3(n - 2)} {4n \times 5(n - 2)} $ $ p = \dfrac{3n(n - 2)}{20n(n - 2)} $ We can cancel the $n - 2$ so long as $n - 2 \neq 0$ Therefore $n \neq 2$ $p = \dfrac{3n \cancel{(n - 2})}{20n \cancel{(n - 2)}} = \dfrac{3n}{20n} = \dfrac{3}{20} $